Threshold Exceedances - Universität zu Köln

The GPD MethodThe Hill Method

Sources

Threshold Exceedances

Moritz Lücke

27. April 2018

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

1 The GPD MethodEstimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

2 The Hill Method

3 Sources

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Excess Distribution

DefinitionLet X be a rv with df F. The excess distribution over the thresholdu has the df

Fupxq “ PpX ´ u ď x |X ą uq “F px ` uq ´ F puq

1´ F puq

for 0 ď x ă XF ´ u.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

The GPD

DefinitionThe General Pareto Distribution (GPD) is given by:

Gξ,βpxq “

#

1´ p1` xξβ q´ 1ξ ξ ‰ 0

1´ expp´ xβ q ξ “ 0

for β ą 0, x ě 0 if ξ ě 0 and 0 ď x ď ´βξ if ξ ă 0.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Abbildung: GPD mit ξ “ ´0.5; 0; 0.5 und β “ 1

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

TheoremWe can find a positive-measurable function βpuq so that

limuÑXF

sup0ďxăXF´u

|Fupxq ´ Gξ,βpuqpxq| “ 0

if and only if F P MDApHξq, ξ P R.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Use of the Theorem

We assume that for some high threshold u, we haveFupxq “ Gξ,βpxq for 0 ď x ă XF ´ u and some ξ P R and β ą 0.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Estimating ξ and β

Given the loss data X1, ...,Xn from F , a random number Nu

will exceed our threshold u.

Relabel these X 11, ...,X1Nu .

We write Y 1j “ X 1j ´ u.We can use the log-likelihood method:

Lpξ, β;Y1, ...,YNuq “

Nuÿ

j“1

lnpgξ,βpYjqq

“ ´Nu lnpβq ´ p1`1ξq

Nuÿ

j“1

lnp1` ξYj

βq,

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Estimating ξ and β

Given the loss data X1, ...,Xn from F , a random number Nu

will exceed our threshold u.Relabel these X 11, ...,X

1Nu .

We write Y 1j “ X 1j ´ u.We can use the log-likelihood method:

Lpξ, β;Y1, ...,YNuq “

Nuÿ

j“1

lnpgξ,βpYjqq

“ ´Nu lnpβq ´ p1`1ξq

Nuÿ

j“1

lnp1` ξYj

βq,

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Estimating ξ and β

Given the loss data X1, ...,Xn from F , a random number Nu

will exceed our threshold u.Relabel these X 11, ...,X

1Nu .

We write Y 1j “ X 1j ´ u.

We can use the log-likelihood method:

Lpξ, β;Y1, ...,YNuq “

Nuÿ

j“1

lnpgξ,βpYjqq

“ ´Nu lnpβq ´ p1`1ξq

Nuÿ

j“1

lnp1` ξYj

βq,

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Estimating ξ and β

Given the loss data X1, ...,Xn from F , a random number Nu

will exceed our threshold u.Relabel these X 11, ...,X

1Nu .

We write Y 1j “ X 1j ´ u.We can use the log-likelihood method:

Lpξ, β;Y1, ...,YNuq “

Nuÿ

j“1

lnpgξ,βpYjqq

“ ´Nu lnpβq ´ p1`1ξq

Nuÿ

j“1

lnp1` ξYj

βq,

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

TheoremWe can find a positive-measurable function βpuq so that

limuÑXF

sup0ďxăXF´u

|Fupxq ´ Gξ,βpuqpxq| “ 0

if and only if F P MDApHξq, ξ P R.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

DefinitionThe mean excess function of an rv X with finite mean is given byepuq “ E pX ´ u|X ą uq.

TheoremUnder the assumption Fupxq “ Gξ,βpxq, it follows that the meanexcess function is linear for all v ą u

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

DefinitionThe mean excess function of an rv X with finite mean is given byepuq “ E pX ´ u|X ą uq.

TheoremUnder the assumption Fupxq “ Gξ,βpxq, it follows that the meanexcess function is linear for all v ą u

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

For positive-valued loss data X1, ...,Xn we estimate the meanexcess function with the sample mean excess function given by

enpvq “

řni“1pXi ´ vqIpXiąvq

řni“1 IpXiąvq

.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

We construct the mean excess plottpXi ,n, enpXi ,nqq : 2 ď i ď nu where Xi denotes the upper ithorder statistic.

If the data support a GDP model over a high threshold, thenthe plot should become increasingly linear for higher values ofv .By this, we can estimate the needed high of our threshold.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

We construct the mean excess plottpXi ,n, enpXi ,nqq : 2 ď i ď nu where Xi denotes the upper ithorder statistic.If the data support a GDP model over a high threshold, thenthe plot should become increasingly linear for higher values ofv .

By this, we can estimate the needed high of our threshold.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

We construct the mean excess plottpXi ,n, enpXi ,nqq : 2 ď i ď nu where Xi denotes the upper ithorder statistic.If the data support a GDP model over a high threshold, thenthe plot should become increasingly linear for higher values ofv .By this, we can estimate the needed high of our threshold.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

modelling Tails and Measures of Tail risk

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

Our goal is to estimate the tail of a underlying loss distribution Fand associated risk measures.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

We have for x ě u

F̄ pxq “ PpX ą uqPpX ą x |X ą uq

“ F̄ puqPpX ´ u ą x ´ u|X ą uq

“ F̄ puqF̄upx ´ uq

“ F̄ puqp1` ξx ´ u

βq´ 1ξ

which gives us a formula for tail probabilities, if F puq is known.If not, we can estimate F̄ puq with the estimator Nu

n .

ñ ˆ̄F pxq “ Nun p1` ξ̂

x´uβ̂q´ 1ξ̂ .

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

We have for x ě u

F̄ pxq “ PpX ą uqPpX ą x |X ą uq

“ F̄ puqPpX ´ u ą x ´ u|X ą uq

“ F̄ puqF̄upx ´ uq

“ F̄ puqp1` ξx ´ u

βq´ 1ξ

which gives us a formula for tail probabilities, if F puq is known.

If not, we can estimate F̄ puq with the estimator Nun .

ñ ˆ̄F pxq “ Nun p1` ξ̂

x´uβ̂q´ 1ξ̂ .

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

We have for x ě u

F̄ pxq “ PpX ą uqPpX ą x |X ą uq

“ F̄ puqPpX ´ u ą x ´ u|X ą uq

“ F̄ puqF̄upx ´ uq

“ F̄ puqp1` ξx ´ u

βq´ 1ξ

which gives us a formula for tail probabilities, if F puq is known.If not, we can estimate F̄ puq with the estimator Nu

n .

ñ ˆ̄F pxq “ Nun p1` ξ̂

x´uβ̂q´ 1ξ̂ .

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

We have for x ě u

F̄ pxq “ PpX ą uqPpX ą x |X ą uq

“ F̄ puqPpX ´ u ą x ´ u|X ą uq

“ F̄ puqF̄upx ´ uq

“ F̄ puqp1` ξx ´ u

βq´ 1ξ

which gives us a formula for tail probabilities, if F puq is known.If not, we can estimate F̄ puq with the estimator Nu

n .

ñ ˆ̄F pxq “ Nun p1` ξ̂

x´uβ̂q´ 1ξ̂ .

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

By inverting the formula, we can obtain a high quantile of theunderlying distribution, which we can interpret as a VaR . Forα ď F puq we have

VaRα “ qαpF q “ u `β

ξpp1´ αF̄ puq

q´ξ ´ 1q.

Assuming ξ ă 1, the associated expected shortfall is given by

ESα “1

1´ α

ż 1

αqxpF qdx “

VaRα1´ ξ

`β ´ ξu

1´ ξ.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

By inverting the formula, we can obtain a high quantile of theunderlying distribution, which we can interpret as a VaR . Forα ď F puq we have

VaRα “ qαpF q “ u `β

ξpp1´ αF̄ puq

q´ξ ´ 1q.

Assuming ξ ă 1, the associated expected shortfall is given by

ESα “1

1´ α

ż 1

αqxpF qdx “

VaRα1´ ξ

`β ´ ξu

1´ ξ.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

By inverting the formula, we can obtain a high quantile of theunderlying distribution, which we can interpret as a VaR . Forα ď F puq we have

VaRα “ qαpF q “ u `β

ξpp1´ αF̄ puq

q´ξ ´ 1q.

Assuming ξ ă 1, the associated expected shortfall is given by

ESα “1

1´ α

ż 1

αqxpF qdx “

VaRα1´ ξ

`β ´ ξu

1´ ξ.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

The Hill Method

Alternative to the GPD Method.

New assumption: F P MDApHξq, ξ ą 0.We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´

1ξ Lpxq for ξ ą 0)

ñ F̄ “ x´αLpxq for a function L P R0 and α “ 1ξ ą 0.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

The Hill Method

Alternative to the GPD Method.New assumption: F P MDApHξq, ξ ą 0.

We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´

1ξ Lpxq for ξ ą 0)

ñ F̄ “ x´αLpxq for a function L P R0 and α “ 1ξ ą 0.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

The Hill Method

Alternative to the GPD Method.New assumption: F P MDApHξq, ξ ą 0.We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´

1ξ Lpxq for ξ ą 0)

ñ F̄ “ x´αLpxq for a function L P R0 and α “ 1ξ ą 0.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

The Hill Method

Alternative to the GPD Method.New assumption: F P MDApHξq, ξ ą 0.We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´

1ξ Lpxq for ξ ą 0)

ñ F̄ “ x´αLpxq for a function L P R0 and α “ 1ξ ą 0.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating α

Given the data X1, ...,Xn, we first build the orderstatisticXn,n ď ... ď X2,n ď X1,n

the hill estimator is then given by

α̂Hk,n “ p

1k

kÿ

j“1

lnpXj ,nq ´ lnpXk,nqq´1

for 2 ď k ď n.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating α

Given the data X1, ...,Xn, we first build the orderstatisticXn,n ď ... ď X2,n ď X1,n

the hill estimator is then given by

α̂Hk,n “ p

1k

kÿ

j“1

lnpXj ,nq ´ lnpXk,nqq´1

for 2 ď k ď n.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating α

Given the data X1, ...,Xn, we first build the orderstatisticXn,n ď ... ď X2,n ď X1,n

the hill estimator is then given by

α̂Hk,n “ p

1k

kÿ

j“1

lnpXj ,nq ´ lnpXk,nqq´1

for 2 ď k ď n.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating α

Given the data X1, ...,Xn, we first build the orderstatisticXn,n ď ... ď X2,n ď X1,n

the hill estimator is then given by

α̂Hk,n “ p

1k

kÿ

j“1

lnpXj ,nq ´ lnpXk,nqq´1

for 2 ď k ď n.

The strategy is to plot Hill estimates for various values of k.This gives the Hill plot (pk, α̂H

k,nq : k “ 2, ..., n). We hope tofind a stable region in the Hill plot.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating α

Given the data X1, ...,Xn, we first build the orderstatisticXn,n ď ... ď X2,n ď X1,n

the hill estimator is then given by

α̂Hk,n “ p

1k

kÿ

j“1

lnpXj ,nq ´ lnpXk,nqq´1

for 2 ď k ď n.The strategy is to plot Hill estimates for various values of k.This gives the Hill plot (pk, α̂H

k,nq : k “ 2, ..., n). We hope tofind a stable region in the Hill plot.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Estimating α

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

hill based tail estimates

We assume a tail of the form F̄ pxq “ Cx´α, x ě u ą 0 forsome high threshold u.

We estimate α by α̂pHqk,n and u by Xk,n

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

hill based tail estimates

We assume a tail of the form F̄ pxq “ Cx´α, x ě u ą 0 forsome high threshold u.

We estimate α by α̂pHqk,n and u by Xk,n

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

estimating C

F̄ puq “ Cu´α̂pHq

k,n

ô C “ uα̂pHq

k,n F̄ puq

The empirical estimator for F̄ puq is kn .

We get the Hill tail estimator

ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

estimating C

F̄ puq “ Cu´α̂pHq

k,n ô C “ uα̂pHq

k,n F̄ puq

The empirical estimator for F̄ puq is kn .

We get the Hill tail estimator

ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

estimating C

F̄ puq “ Cu´α̂pHq

k,n ô C “ uα̂pHq

k,n F̄ puq

The empirical estimator for F̄ puq is kn .

We get the Hill tail estimator

ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

estimating C

F̄ puq “ Cu´α̂pHq

k,n ô C “ uα̂pHq

k,n F̄ puq

The empirical estimator for F̄ puq is kn .

We get the Hill tail estimator

ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.

Moritz Lücke Threshold Exceedances

The GPD MethodThe Hill Method

Sources

Sources

[1] A.McNeil R.Frey P.Embrechts. Quantitative Risk Management:Concepts, Techniques and Tools. Princeton Series in Finance, 2015.

Moritz Lücke Threshold Exceedances